VON NEUMANN, John. & GOLDSTINE, Herman H. Numerical Inverting of Matrices of High Order.
The American Mathematical Society, 1947. First edition.

Rare offprint of one of von Neumann’s major papers. “The 1947 paper by John von Neumann and Herman Goldstine, ‘Numerical Inverting of Matrices of High Order’ (Bulletin of the AMS, Nov. 1947), is considered as the birth certificate of numerical analysis. Since its publication, the evolution of this domain has been enormous.” (Bultheel & Cools: The Birth of Numerical Analysis).

“Just when modern computers were being invented (those digital, electronic, and programmable), John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did concerning the accuracy of mechanized Gaussian elimination - but their paper was about more than that. Von Neumann and Goldstine described what they surmised would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them.” (John von Neumann's Analysis of Gaussian Elimination: the Founding of Modern Numerical Analysis, Lecture at ICME, Stanford, January 2007).

“The concluding Chapter 7 interprets the rounding error analysis. Von Neumann asked his readers to continue from his residual bound ‘several different ways’. He guided his readers by explaining the appropriate conclusion shows the computed result is exact for some perturbation of the initial data. This is the ‘backward’ interpretation which many people think von Neumann did not understand. The paper closes by evaluating the residual bound for ‘random’ matrices, and by counting arithmetic operations.
In sum, von Neumann’s paper contains much that is unappreciated or at least unattributed to him. The contents are so familiar, it is easy to forget von Neumann is not repeating what everyone knows. He anticipated many of the developments in the field he originated, and his theorems on the accuracy of Gaussian elimination have not been encompassed in half a century. The paper is among von Neumann's many firsts in computer science. It is the first paper in modern numerical analysis, and the most recent by a person of von Neumann's genius.” (Kees Vuik, Birthday of Modern Numerical Analysis).

Hook & Norman, Origins of Cyberspace, no. 957; Goldstine, The Computer from Pascal to von Neumann, pp.290-91; Nash, A History of Scientific Computing, p.23; DSB, XIV, p.92.

Offprint from Bulletin of the American Mathematical Society, vol. 53, no. 11, pp. 1021-1099. 8vo: 241 x 152 mm. Original green printed wrappers. Provenance: With the rubber stamp of statistician, and specialist in numerical methods, Dr. Clifford J. Maloney to the front and rear wrapper. Very fine.